The Difference a Different Voting System Makes

Alex Tabarrok

Different Voting Systems

Individual Rankings

(Inputs)B D

AC C

CA B

DD A

B

Voting System

(Aggregation Mechanism)

Election Outcome

(Global Ranking)BDAC

What is the Will of the Voters?

A Simple Election Example

Number of Voters → 39 24 37

1st A C B

2nd C B C

3rd B A A

• Plurality Rule: A>B>C

• Borda Count: C>B>A

• “7,1,0” Count: B>A>C

Mathematics of Point Score Voting Systems Assume n=3 candidates then we can write the

plurality rule system as (1,0,0) and the Borda Count as (2,1,0).

Clearly, (10,0,0) is equivalent to plurality rule. It's also true although a little bit more difficult to see that (5,3,1) is equivalent to the Borda Count.

Any point score voting system can be converted into a standardized point score system denoted (1-s,s,0), where s∈[0, 1/2].

The standardized plurality rule system is s= The standardized Borda Count is s=

1

0

0

2

1

0

Plurality Rule

BordaCount

1-s

s

0

Standardized Point Score System

01

3

?

?

Mathematics of Point Score Voting Systems

A voter may rank three candidates in any one of six possible ways.

The vote matrix can be read in two ways. Reading down a particular column we

see the number of points given to each candidate from a voter with the ranking indicated by that column.

Reading across the rows we see where a candidate's votes come from.

We will write the number of voters with ranking (1) ABC as p₁ the number of voters with ranking (2) ACB as p₂ and so forth up until p₆. We can place all this information in matrix form by multiplying the vote matrix with the voter type matrix.

ABC ACB CAB CBA BCA BACA 1-s 1-s s 0 0 sB s 0 0 s 1-s 1-sC 0 s 1-s 1-s s 0

p₁ (ABC)p₂

(ACB)p₃

(CAB)p₄

(CBA)p₅

(BCA)p₆

(BAC)

1-s 1-s s 0 0 ss 0 0 s 1-s 1-s0 s 1-s 1-s S 0

=

A’s tallyB’s

tallyC’s

tally

Familiar Examples

0390

24370

1 1 0 0 0 00 0 0 0 1 10 0 1 1 0 0 =

393724

0390

24370

2/3 2/3 1/3 0 0 1/31/3 0 0 1/3 2/3 2/30 1/3 2/3 2/3 1/3 0

=26

32.6641.33

Plurality Rule

Borda Count

A Simple Election Example

39 24 37

1st A C B

2nd C B C

3rd B A A

p₁ =0(ABC)

p₂=39(ACB)

p₃=0(CAB)

p₄=24(CBA)

p₅=37(BCA)

p₆=0(BAC)

The Representation Triangle and the Procedure Line

p₁+p₂+(-p₁-p₂+p₃+p₆) s∗p₆+p₅+(p₄-p₅+p₁-p₆) s∗p₃+p₄+(p₂-p₃-p₄+p₅) s∗

1-s 1-s s 0 0 ss 0 0 s 1-s 1-s0 s 1-s 1-s s 0 =

0.5 1A

0.5

1B

1) ABC

2)ACB 3)CAB

5)BCA 6)BAC

C

4)CBA

p₁p₂p₃p₄p₅p₆

• Interpret p₁…p6 as shares of each type of voter then the tallies are vote shares.

• Vote shares must sum to 100% so one of the equations is redundant. Thus we can graph in 2-dimensions.

A Simple Election Example

39 24 37

1st A C B

2nd C B C

3rd B A A

Plurality Rule: A>B>C

Borda Count: C>B>A

“7,1,0” Count: B>A>C

0390

24370

39-39 s∗37-13 s∗24+52 s∗

One Ranking Many Outcomes

Maximum Outcomes from One Ranking

Candidates OutcomesMax. Outcomes from 1 Ranking

Max. Outcomes from 1 Ranking_____________Total Outcomes

2 2 1 0.53 6 4 0.664 24 18 0.755 120 96 0.86 720 600 0.8337 5040 4320 0.8578 40,320 35,280 0.8759 362,880 322,560 0.888

10 3,628,800 3,265,920 0.9n n! n!-(n-1)! 1-(1/n)

Source: Saari (1992)

0.5 1A

0.5

1B

1) ABC

2)ACB 3)CAB

5)BCA 6)BAC

C

4)CBA

Intuition for these results comes from the geometry of the procedure line extended to higher dimensions.

Many Outcomes is the Norm

For even small electorates (say 50 or more) and 3 candidates a single profile generates: 7 different rankings (including ties) about 6.7 percent of the time 5 different rankings 18.6 percent of the time, 3 different rankings 41.3 percent of the time a single ranking 33.3 percent of the time.

A single profile, therefore, generates more than one ranking 66 percent of the time.

As the number of candidates increases the probability that all positional voting systems agree on the winner (K=1) quickly goes to zero.

The 1992 Election

0.5 1 Clinton

0.5

1

Bush

1

23

4

5 6

Perot

Pluralityy Rule

Borda Count

Anti-Plurality Rule

• Multiple outcomes from the same profile are not always the case.• In 1992, conservative commentators emphasized President Clinton's failure to receive more than 50% of the vote and thus his failure, in their minds, to achieve a "mandate.“• An analysis of voter preferences, however, reveals the surprising fact that Clinton would have won under any point-score voting system!

President Perot and Approval Voting

0.5 1 Clinton

0.5

1

Bush

1

23

4

5 6

Perot

Pluralityy Rule

Borda Count

Anti-Plurality Rule

• Approval voting is an increasing popular system where each voter can approve of as many candidates as he or she likes.• e.g. if there are 5 candidates the voter could approve 1,2,3, or 4 of them.• Approval voting vastly increases what can happen. Note, for example, that with candidates under approval voting each voter has the option of using plurality rule or anti-plurality rule!• What could have happened in 1992? Anything!

Voting

Group choice is not at all like individual choice.

Groups will always choose in ways that would appear irrational if chosen by an individual.

The voting system determines the outcome of an election at least as much as do preferences.

Voting does not represent the “will of the voters.”

The idea of a group will is incoherent.

In Defense of Democracy

Nobel prize winner Amartya Sen has argued that: “No famine has ever taken

place in the history of the world in a functioning democracy.”

“Democracies have to win elections and face public criticism, and have strong incentive to undertake measures to avert famines and other catastrophes.”

Democracy and Famines

The Democratic and Capitalist Peace

Democratic Peace – democracies rarely go to war against one another.

Capitalist Peace – trading countries, countries with private property and capitalist economies rarely go to war against one another.

Democratic and capitalist peace are strongly supported in the data and a consensus has developed in the International Relations literature but less consensus on why.

Economic Freedom, Democracy and Living Standards

The Democratic/Capitalist Peace

Democracies

Free Economies

10

20

30

40

50

60

Fre

e E

cono

mie

s

0

10

20

30

40

50

60

70

Num

ber

of D

emo

crac

ies

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

2010

Year

Source: Polity IV, Democracy measured as democ>=8.Economic Freedom of the World 2010, measured as chained summary index>7

1900-2009The Growth of Democracy and Economic Freedom

International Conflict is Down

So What is Democracy Good For?

Democracies don’t kill their own citizens or let them starve. Democracy is compatible with economic freedom ->

democratic/capitalist peace. Democracies avoid some very bad possibilities. The threat of throwing politicians out of office is a constraint on what

can happen in a democracy. Dictatorships and oligarchies need only not abuse a minority – in a

democracy the standard is higher. Democracy, however, is not good at representing the will of the

voters and in general we should not expect democracy to be a good way of making decisions.

Democracy should be seen as a way of limiting or constraining government.